          Astron. Astrophys. 363, 1186-1194 (2000)

## 3. The numerical code

The code used to produce the results (Lare2d ) was developed at St. Andrews University by Arber et al. (2000). Lare2d is a numerical code that operates by taking a Lagrangian predictor-corrector time step and after each Lagrangian step all variables are conservatively remapped back onto the original Eulerian grid using Van Leer gradient limiters. Results are obtained by initializing the code with the unperturbed density where is a free parameter used to fix the scale length of the density inhomogeneity. An example with is shown in Fig. 1. This is the value of used throughout unless otherwise explicitly stated. Note, that the position of the highest gradient in the Alfvén speed ( ) is shifted toward the positive x with respect to the positioning of the highest gradient in the density ( ). Fig. 1. Profiles of the unperturbed density (solid curve ) and Alfvén speed (dashed curve ) for the inhomogeneity parameter .

The boundary condition in the x direction is that all quantities have zero gradient. The x boundaries are far away from the maximum background density change at , so that they do not influence the physics.

In this geometry linearly polarized Alfvén waves perturb the y components of the magnetic field and velocity ( and ). This is implemented numerically in two different ways:

• Firstly, the z direction is treated as periodic and an Alfvén wave with amplitude 0.001 is initialized so that two wavelengths fit into the numerical domain. The wave is planar at with wave-fronts parallel to the x axis. (Sect. 4)

• Secondly, a sinusoidal Alfvén wave is driven at the boundary: . Again the amplitude is and driving frequency is . The driven Alfvén wave propagates in the z direction and as soon as the wave reaches the upper boundary the simulation is stopped. (Sect. 5)

A plasma- of 0.01 is used, to correspond to the plasma in the solar corona. Unless otherwise stated, the initialization described in this section has been used for all the numerical simulations presented in this paper.

Previous phase mixing calculations (De Moortel et al. 1999) have used scale lengths and speeds estimated for coronal plumes. One of the aims of this work is to estimate the relative importance of nonlinear generation of fast modes, as opposed to classical phase mixing, as a mechanism for dissipating Alfvén wave energy. A typical plume in this paper is assumed to have a number density of , , and a background density of . An increase in density by a factor of 4 is assumed to take place over a distance of . ( .) For these parameters a one minute period Alfvén wave would have a wavelength , while five minute oscillations would have wavelengths . Furthermore for the one minute oscillations it is expected (De Moortel et al. 1999) that phase mixing will become prevalent at around 1.5 solar radii. For coronal plumes this suggests that the Alfvén wave is of the order of 10 full wavelengths out of phase when classical phase mixing is significant. Simulations in this work are therefore for density scale lengths less than the Alfvén wavelength, with a density increase by a factor of four, and run for sufficient time that the waves go out of phase by 10 wavelengths across the density ramp.    © European Southern Observatory (ESO) 2000

Online publication: December 5, 2000 